The main magnetic field of the Earth is a complex phenomenon. To understand its origins in the fluid of the Earth's core, and how it changes in time requires a variety of mathematical and physical tools. This book presents the foundations of geomagnetism, in detail and developed from first principles. The book is based on George Backus' courses for graduate students at the University of California, San Diego. The material is mathematically rigorous, but is logically developed and has consistent notation, making it accessible to a broad range of readers. The book starts with an overview of the phenomena of interest in geomagnetism, and then goes on to deal with the phenomena in detail, building the necessary techniques in a thorough and consistent manner. Students and researchers will find this book to be an invaluable resource in the appreciation of the mathematical and physical foundations of geomagnetism.

The frozen-flux hypothesis for the Earth's liquid core assumes that convective terms dominate diffusive terms in the induction equation governing the behaviour of the magnetic field at the surface of the core. While highly plausible on the basis of estimates of physical parameters, the hypothesis has been questioned in recent work by Bloxham, Gubbins & Jackson (1989) who find it to be inconsistent with their field models for most of the century. To study this question we improve the method of Constable, Parker & Stark (1993), which tests the consistency of magnetic observations with the hypothesis by constructing simple, flux-conserving core-field models fitting the data at pairs of epochs. We introduce a new approach that fixes the patch configurations at each of the two epochs before inversion, so that each configuration is consistent with its respective data set but possesses the same patch topology. We expand upon the inversion algorithm, using quadratic programming to maintain the proper flux sign within patches; the modelling calculations are also extended to include data types that depend non-linearly on the model. Every test of a hypothesis depends on the characterization of the observational uncertainties; we undertake a thorough review of this question. For main-field models, the primary source of uncertainty comes from the crustal field. We base our analysis on one of Jackson's (1994) statistical models of the crustal magnetization, adjusted to bring it into better conformity with our data set. The noise model permits us to take into account the correlations between the measurements and requires that a different weighting be given to horizontal and vertical components. It also indicates that the observations should be fit more closely than has been the practice heretofore. We apply the revised method to Magsat data from 1980 and survey and observatory data from 1915.5, two data sets believed to be particularly difficult to reconcile with the frozen-flux hypothesis. We compute a pair of simple, flux-conserving models that fit the averaged data from each epoch. We therefore conclude that present knowledge of the geomagnetic fields of 1980 and 1915.5 is consistent with the frozen-flux hypothesis.